We study existence and regularity of distributional solutions for a class of nonlinear parabolic problems. The equations we consider have a quasi-linear diffusion operator and a lower-order term, which may grow quadratically in the gradient and may have a very fast growth (for instance, exponential) with respect to the solution. The model problem we refer to is the following \begin{equation} \begin{cases} u_t -\Delta u = \beta(u) |{\nabla} u|^2 +f(x, t), & \ \text{in}\quad \Omega\times (0, T);\\[0.1cm] u(x,t)=0, & \ \text{on}\quad \partial\Omega\times (0, T);\\[0.1cm] u(x, 0)=u_0(x), & \ \text{in}\quad \Omega; \end{cases} \tag*{(0.1)} \label{modello} \end{equation} with $\Omega \subset \mathbb R^N$ a bounded open set, $T>0$, and $\beta(u)\sim e^{|u|}$; as far as the data are concerned, we assume $\exp(\exp(|u_0|))\in L^2({\Omega})$, and $f \in X(0, T; Y({\Omega}))$, where $X, Y$ are Orlicz spaces of logarithmic and exponential type, respectively. We also study a semilinear problem having a superlinear reaction term, a problem that is linked with problem (0.1) by a change of unknown (see (1.3) below). Likewise, we deal with some other related problems, which include a gradient term and a reaction term together.

We prove existence, uniqueness and Schauder estimates for the degenerate elliptic and parabolic equations (E) and (NHCP) in $\mathbb R^N$ associated to the degenerate Kolmogorov operator (K) defined below.

We prove that the first eigenvalue of a vector-valued $p$-Laplacian problem is equal to the first eigenvalue of the corresponding scalar $p$-Laplacian, and that the components of its first eigenvectors are merely copies of the first eigenfunction of the scalar problem. We also show variants of this result for some other homogeneous vector-valued problems.

The study of the maximum recoverable work is given in terms of Fourier-transformed quantities for incompressible viscoelastic fluids. This maximum is related to the minimum free energy of the system, whose expression is found as a function of the minimal state. Finally, these results are developed for a discrete spectrum model.

We consider a second-order weakly hyperbolic equation with time and space depending coefficients. We suppose the coefficients to have globally a H\"older type behavior and locally a blow up of the first derivative at some time. We show that the Cauchy problem for such an equation is well posed in Gevrey classes $G^\sigma$; the upper bound for the Gevrey index $\sigma$ depends only on the dominant between the local and the global condition.